A kneading map of chaotic switching oscillations in a Kerr cavity with two interacting light fields

Abstract

Optical systems that combine nonlinearity with coupling between various subsystems offer a flexible platform for observing a diverse range of nonlinear dynamics. Furthermore, engineering tolerances are such that the subsystems can be identical to within a fraction of the wavelength of light; hence, such coupled systems inherently have a natural symmetry that can lead to either delocalization or symmetry breaking. We consider here an optical Kerr cavity that supports two interacting electric fields, generated by two symmetric input beams. Mathematically, this system is modeled by a four-dimensional $\mathbb{Z}_2$-equivariant vector field with the strength and detuning of the input light as control parameters. Previous research has shown that complex switching dynamics are observed both experimentally and numerically across a wide range of parameter values. Here, we show that particular switching patterns are created at specific global bifurcations through either delocalization or symmetry breaking of a chaotic attractor. We find that the system exhibits infinitely many of these global bifurcations, which are organized by $\mathbb{Z}_2$-equivariant codimension-two Belyakov transitions. We investigate these switching dynamics by means of the continuation of global bifurcations in combination with the computation of kneading invariants and Lyapunov exponents. In this way, we provide a comprehensive picture of the interplay between different switching patterns of periodic orbits and chaotic attractors.

Figures (8)

• Figure 1: One-parameter bifurcation diagram of system \ref{['eq1']} in the detuning $\Delta$, for $F=4.0$. Panel (a1) is an overall view and panel (a2) shows an enlargement in the colored frame. Stable and unstable symmetric solutions are represented by solid and dashed blue curves, respectively, and asymmetric states are shown in purple. Branches of periodic orbits (green) are represented here by the sum of the squared maxima of their real and imaginary parts, i.e., $P_{1,2}= \max(X_{1,2})^2 + \max(Y_{1,2})^2$. (c1)--(e1) Phase portraits with stable periodic orbits and the positive branch $W_+^u(p)$, (c2)--(e2) associated temporal traces, and (c3)--(e3) power spectra of the output intensities $P_1$ and $P_2$, for $(F,\Delta)=(4.0, 6.8053)$, $(F,\Delta)=(4.0, 6.8054)$, and $(F,\Delta)=(4.0, 7.16)$, respectively.
• Figure 2: (a) Phase diagram in the $(F, \Delta)$-plane showing a selection of bifurcation curves of system \ref{['eq1']}. Shown are the curve of Hopf $\mathbf{H}$ (green), Shilnikov $\text{HOM}_p$ (dark blue, light blue and blue), pitchfork $\mathbf{SB}$ (fuchsia), and period-doubling $\mathbf{PD}_{1,2}$ (teal and dark green) bifurcations. The large and small insets show enlargements; in the colored frame in light brown and around the orange star, respectively. (b1)--(c1) Phase portraits with chaotic attractors and the positive branch $W_+^u(p)$, (b2)--(c2) associated temporal traces, and (b3)--(c3) power spectra of $P_1$ and $P_2$, for $(F, \Delta)=(6.2446,7.1740)$ and $(F, \Delta)= (6.2446,7.1753)$, as indicated in the small inset of panel (a).
• Figure 3: Kneading sequence generated by the temporal trace of $P_{\rm diff}=P_2 - P_1$ along the positive branch $W^u_+(p)$ for $(F,\Delta)=(7.78103,8.08573)$; the excluded range $[x^-,x^+]$ is shaded.
• Figure 4: (a1)--(c1) Three different Shilnikov homoclinic orbits formed by $W_+^u(p)$, and (a2)--(c2) corresponding temporal profiles, for $F=7.6$ and $\Delta=7.837$, $\Delta=7.897$ and $\Delta=7.624$, respectively.
• Figure 5: Kneading maps of (a) $I_2$, (b) of $I_3$, and (c) of $I_5$ in the $(\widetilde{F},\widetilde{\Delta})$-plane near the Belyakov transition point $\textbf{BV}$ of system \ref{['eq1']}, shown together with computed curves of Shilnikov bifurcations with up to four loops. Coloring distinguishes regions with different switching patterns, as indicated by the color bars.